Here is a collection of documents I have made over the years. I cannot claim that they are complete, neat, or even always correct, but they helped get me through a maths degree, and cover lots of interesting concepts.

Here is my masters dissertation:

The Inverse Galios Problem – The Rigidity Method  – this gives an introduction to the inverse Galois problem, develops the relevant theory required to understand the rigidity method, a method that has been used to great effect in this area, and proves the rigitidy criterion. This takes up about the first half of the paper. The second half is a collection of numerous applications of this method culminating in one of the more celebrated results in the area; proof that the monster group can be realised as a Galois group!

Here is a collection of some notes I made for several courses during my undergrad at Imperial. These were typed up during the lectures so expect a few errors.

Riemann Geometry – the contents seems to have fallen off this document. It contains

1. An introduction to manifolds
2. Riemannian metrics and manifolds
3. Geodesics
4. Curvature
5. Jacobi fields
6. Completeness
7. Constant curvature

Groups and Representation Theory – this is very rough around the edges, with annotations and questions I had during the lectures dotted around. It remains a beautiful course however.

Elliptic Curves – covers the main concepts and results in this area. Introduces the p-adic numbers and considers elliptic curves over $\mathbb{Q}_p$